The problem of general Radon representation for an arbitrary Hausdorff space

V. K. Zakharov ; A. V. Mikhalev

Izvestiya. Mathematics , Tome 63 (1999) no. 5, pp. 881-921.

Résumé

After the fundamental work of Riesz, Radon and Hausdorff in the period 1909–1914, the following problem of general Radon representation emerged: for any Hausdorff space find the space of linear functionals that are integrally representable by Radon measures. In the early 1950s, a partial solution of this problem (the bijective version) for locally compact spaces was obtained by Halmos, Hewitt, Edwards, Bourbaki and others. For bounded Radon measures on a Tychonoff space, the problem of isomorphic Radon representation was solved in 1956 by Prokhorov. In this paper we give a possible solution of the problem of general Radon representation. To do this, we use the family of metasemicontinuous functions with compact support and the class of thin functionals. We present bijective and isomorphic versions of the solution (Theorems 1 and 2 of § 2.5). To get the isomorphic version, we introduce the family of Radon bimeasures.

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